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In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables (,) = + +,where a, b, c are the coefficients.When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form.
In "twos in", binary quadratic forms are of the form ax 2 + 2bxy + cy 2, represented by the symmetric matrix This is the convention Gauss uses in Disquisitiones Arithmeticae. In "twos out", binary quadratic forms are of the form ax 2 + bxy + cy 2 , represented by the symmetric matrix ( a b / 2 b / 2 c ) . {\displaystyle {\begin{pmatrix}a&b/2\\b ...
In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae, [1] a textbook on number theory published in 1801, in Articles 234 - 244.
These three quadratic forms all have the same discriminant and Manjul Bhargava proved that their composition in the sense of Gauss [3] is the identity element in the associated group of equivalence classes of primitive binary quadratic forms.
The first of these is the class number formulae for binary quadratic forms. The second is a proof that arithmetic progressions contains an infinite number of primes (known as Dirichlet's theorem); this proof introduces Dirichlet L-series. These results are important milestones in the development of analytic number theory.
For binary quadratic forms there is a group structure on the set C of equivalence classes of forms with given discriminant. The genera are defined by the generic characters . The principal genus, the genus containing the principal form, is precisely the subgroup C 2 and the genera are the cosets of C 2 : so in this case all genera contain the ...
The Disquisitiones include the Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares , he proves the triangular case of the Fermat polygonal number theorem for n = 3. [ 116 ]
Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. There remain some unsolved problems. The class number problem is particularly important.